Papers by Ludovick Bouthat
arXiv (Cornell University), Jun 8, 2023
The objective of the present paper is to establish three Hardy-type inequalities in which the ari... more The objective of the present paper is to establish three Hardy-type inequalities in which the arithmetic mean over a sequence of non-negative real numbers is replaced by some weighted arithmetic mean over some nested subsets of the given sequence of numbers. One of these inequalities stems from a calculation in a paper of Bouthat and Mashreghi on semi-infinite matrices.
arXiv (Cornell University), Jun 8, 2023
In a celebrated paper of Marcus and Ree (1959), it was shown that if A = [a ij ] is an n × n doub... more In a celebrated paper of Marcus and Ree (1959), it was shown that if A = [a ij ] is an n × n doubly stochastic matrix, then there is a permutation σ ∈ Sn such that n i,j=1 a 2 i,j ≤ n i=1 a i,σ(i). Erdős asked for which doubly stochastic matrices the inequality is saturated. Although Marcus and Ree provided some insight for the set of solutions, the question appears to have fallen into oblivion. Our goal is to provide a complete answer in the particular, yet non-trivial, case when n = 3.
Operator theory, Nov 4, 2022
arXiv: Functional Analysis, Sep 20, 2021
In this note we study the induced p-norm of circulant matrices A(n, ±a, b), acting as operators o... more In this note we study the induced p-norm of circulant matrices A(n, ±a, b), acting as operators on the Euclidean space R n. For circulant matrices whose entries are nonnegative real numbers, in particular for A(n, a, b), we provide an explicit formula for the p-norm, 1 ≤ p ≤ ∞. The calculation for A(n, −a, b) is more complex. The 2-norm is precisely determined. As for the other values of p, two different categories of upper and lower bounds are obtained. These bounds are optimal at the end points (i.e. p = 1 and p = ∞) as well as at p = 2.
The L-matrix A_s=[1/(n+s)] was introduced in <cit.>. As a surprising property, we showed th... more The L-matrix A_s=[1/(n+s)] was introduced in <cit.>. As a surprising property, we showed that its 2-norm is constant for s ≥ s_0, where the critical point s_0 is unknown but relies in the interval (1/4,1/2). In this note, using some delicate calculations we sharpen this result by improving the upper and lower bounds of the interval surrounding s_0. Moreover, we show that the same property persists for the p-norm of A_s matrices. We also obtain the 2-norm of a family of C-matrices with lacunary sequences.
Operators and Matrices
We show that an L-matrices A = [a n ] , with lacunary coefficients (a n) is a bounded operator on... more We show that an L-matrices A = [a n ] , with lacunary coefficients (a n) is a bounded operator on 2 , provided that (a n) satisfy an explicit decay rate. Moreover, by a concrete example, we see that the decay restriction is optimal. The extension to operators on p spaces, for p > 1 , is also discussed.
In this note we study the induced p-norm of circulant matrices A(n,± a, b), acting as operators o... more In this note we study the induced p-norm of circulant matrices A(n,± a, b), acting as operators on the Euclidean space ℝ^n. For circulant matrices whose entries are nonnegative real numbers, in particular for A(n,a,b), we provide an explicit formula for the p-norm, 1 ≤ p ≤∞. The calculation for A(n,-a,b) is more complex. The 2-norm is precisely determined. As for the other values of p, two different categories of upper and lower bounds are obtained. These bounds are optimal at the end points (i.e. p=1 and p = ∞) as well as at p=2.
The L-matrix As = [1/(n+s)] was introduced in [1]. As a surprising property, we showed that its 2... more The L-matrix As = [1/(n+s)] was introduced in [1]. As a surprising property, we showed that its 2-norm is constant for s ≥ s0, where the critical point s0 is unknown but relies in the interval (1/4, 1/2). In this note, using some delicate calculations we sharpen this result by improving the upper and lower bounds of the interval surrounding s0. Moreover, we show that the same property persists for the p-norm of As matrices. We also obtain the 2-norm of a family of C-matrices with lacunary sequences.
Linear Algebra and its Applications, 2021
Ludovick Bouthata, Apoorva Khareb,c, Javad Mashreghia, and Frédéric Morneau-Guérin∗,a,d ∗ Corresp... more Ludovick Bouthata, Apoorva Khareb,c, Javad Mashreghia, and Frédéric Morneau-Guérin∗,a,d ∗ Corresponding author. Département de mathématiques et de statistique, Université Laval, 1045, avenue de la Médecine, Québec (Québec), G1V 0A6, Canada. b Department of Mathematics, Indian Institute of Science, Bengaluru – 560012, India. c Analysis and Probability Research Group, Bengaluru – 560012, India. Département Éducation, Université TÉLUQ, 455 rue du Parvis, Québec (Québec), G1K 9H6, Canada.
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Papers by Ludovick Bouthat